Hostname: page-component-78c5997874-s2hrs Total loading time: 0 Render date: 2024-11-14T23:22:44.225Z Has data issue: false hasContentIssue false
  • English
  • Français

Level crossings of a random trigonometric polynomial with dependent coefficients

Part of: Stochastic processes

Published online by Cambridge University Press:  09 April 2009

K. Farahmand
K. Farahmand
Affiliation:
Department of Mathematics, University of Ulster, Jordanstown, Co Antrim, BT37 OQB, United Kingdom
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

This paper provides an asymptotic estimate for the expected number of K-level crossings of the random trigonometric polynomial g1 cos x + g2 cos 2x+ … + gn cos nx where gj (j = 1, 2, …, n) are dependent normally distributed random variables with mean zero and variance one. The two cases of ρjr, the correlation coeffiecient between the j-th and r-th coefficients, being either (i) constant, or (ii) ρ∣j−r∣ρ, j ≠ r, 0 < ρ < 1, are considered. It is shown that the previous result for ρjr = 0 still remains valid for both cases.

Keywords

number of real rootsKac-Rice formularandom trigonometric polynomial.

MSC classification

Secondary: 60G99: None of the above, but in this section
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 58 , Issue 1 , February 1995 , pp. 39 - 46
Copyright
Copyright © Australian Mathematical Society 1995

References

[1]Cramér, H. and Leadbetter, M. R., Stationary and related stochastic processes (Wiley, New York, 1967).Google Scholar
[2]Dunnage, J. E. A., ‘The number of real zeros of a random trigonometric polynomial’, Proc. London Math. Soc. 16 (1966), 53–84.CrossRefGoogle Scholar
[3]Farahmand, K., ‘On the average number of level crossings of a random polynomial’, Ann. Probab. 18 (1990), 1403–1409.CrossRefGoogle Scholar
[4]Farahmand, K., ‘Level crossings of a random trigonometric polynomial’, Proc. Amer. Math. Soc. 111 (1991), 551–557.CrossRefGoogle Scholar
[5]Renganathan, N. and Sambandham, M., ‘On the average number of real zeros of a tandom trigonometric polynomial with dependent coefficients’, Indian J. Pure Appl. Math. 15 (1984), 951–956.Google Scholar
[6]Rice, S. O., ‘Mathematical theory of random noise’, Bell System Tech. 25 (1945), 46–156.CrossRefGoogle Scholar
[7]Rudin, W., Real and complex analysis, 2nd edition (McGraw-Hill, New York, 1974).Google Scholar
[8]Sambandham, M., ‘On random trigonometric polynomial’, Indian J. Pure Appl. Math. 7 (1976), 841–849.Google Scholar
[9]Titchmarsh, E. C., The theory of functions, 2nd edition (Oxford University Press, Oxford, 1939).Google Scholar